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Negative probability
Andreas Blass, Yuri Gurevich
TL;DR
The paper asks how a physically meaningful joint description can be formed when position and momentum cannot have an ordinary simultaneous distribution and negative values appear. It develops a characterization based on marginal distributions and presents a simpler direct proof of uniqueness. The key result is that Wigner’s function is the unique quasi-distribution with correct marginals for position, momentum, and all their linear combinations.
Problem
Position and momentum have individual probability distributions, but their joint distribution has no physical sense, while the meaning of negative probabilities remains unclear.
Method
The paper characterizes Wigner’s function through correct marginals for position, momentum, and their linear combinations, and presents a direct uniqueness proof.
Results
Wigner’s function is the unique phase-space quasi-distribution yielding correct marginals for position, momentum, and all their linear combinations.
Takeaways & Limitations
Considering all linear combinations resolves the non-uniqueness left by matching only position and momentum marginals.
Takeaways & Limitations
The paper gives only a cursory discussion of Feynman’s discrete four-outcome quasi-distribution and leaves the relation between negativity and contextuality beyond its scope.
Abstract
from arXiv · showhide
This article was written for the Logic in Computer Science column in the February 2015 issue of the Bulletin of the European Association for Theoretical Computer Science. The intended audience is general computer science audience. The uncertainty principle asserts a limit to the precision with which position x and momentum p of a particle can be known simultaneously. You may know the probability distributions of x and p individually but the joint distribution makes no physical sense. Yet Wigner exhibited such a joint distribution f(x,p). There was, however, a little trouble with it: some of its values were negative. Nevertheless Wigner's discovery attracted attention and found applications. There are other joint distribution, all with negative values, which produce the correct marginal distributions of x and p. But only Wigner's distribution produces the correct marginal distributions for all linear combinations of position and momentum. We offer a simple proof of the uniqueness and discuss related issues.
1. Introduction
The paper examines why negative quasi-probabilities arise in quantum mechanics and develops an objective characterization of Wigner’s function. Its central result is that Wigner’s function is uniquely determined by correct marginals for position, momentum, and all their linear combinations.
- Motivation: Negative probabilities lack a remotely natural interpretation and conflict with the frequentist interpretation when every outcome is observable.Unobservable quantum outcomes weaken, but do not resolve, the frequentist objection.
- Motivation: Quantum mechanics permits negative quasi-probabilities because simultaneous position–momentum joint probabilities have no physical sense, although their individual distributions are known.Wigner’s function supplies such a joint object, but some values are negative and it cannot be interpreted as an ordinary simultaneous probability.
- Main result: Wigner’s function is the unique phase-space quasi-distribution with correct marginals for position, momentum, and every linear combination of them.The paper calls this Proposition 1, Wigner Uniqueness.
- Problem: Position and momentum marginals alone do not determine Wigner’s quasi-distribution, because modifications can preserve both marginals.The paper gives rectangle-based and smooth modifications as examples.
- Approach: The authors present a proof they judge simpler and more direct than earlier proofs, and also establish Moyal’s characterization.The linear-combination idea came from Wigner’s observation that projections preserve expectations and yield correct marginals.
- Scope: The paper treats relations to earlier characterizations and to contextuality, but leaves the discrete history and negativity–contextuality relation outside its scope.Its discussion of Feynman’s four-outcome quasi-distribution is explicitly cursory.
2. Preliminaries
This section introduces Fourier transforms, generalized functions, and exponential operators as analytical tools, with convergence handled through L2 extensions. It shows that exponentiating the derivative operator yields the shift operator on L2(R).
- Fourier-transform arguments use real variables whose product ξx is dimensionless in applications.
- The Fourier transform is treated rigorously on nice functions in L2(R), then extended to all of L2 by isometry and density.
- Dirac’s δ-function and certain divergent integrals are handled as generalized functions.
- The exponential of an operator is introduced through its power series, with the derivative operator producing Taylor shifts on suitable functions.
- Gaussian functions provide a dense subspace of L2(R), allowing the shift f(x) 7→ f(x + a) to extend uniquely and continuously to all of L2(R).
- The exponential e^O may be only partially defined, depending on where the operators O_k are defined.
3. Joint-to-Marginal Lemma
The joint-to-marginal lemma characterizes the marginal of z = ax + bp through the Fourier transform of the joint distribution. This equivalence leads to a pointwise characterization of the joint distribution’s transform.
- For z = ax + bp with a and b not both zero, its marginal distribution is defined from the joint distribution on R2.
- Differential forms simplify the change-of-variables calculation, although the same result follows from a Jacobian determinant.
- The probability assigned to every interval u ≤ z ≤ v is obtained from the corresponding marginal integral.
- Lemma 2 states that being the marginal of z = ax + bp is equivalent to a Fourier-transform identity relating g and f.
- Corollary 3 characterizes the Fourier transform of f at every nonzero point (α, β) using the marginal transform for a suitable linear combination.
4. Wigner uniqueness
The paper proves Wigner’s uniqueness proposition by combining quantum measurement marginals with Fourier analysis. The proof reconstructs the quasi-distribution and identifies it with Wigner’s formula.
- The uniqueness proof is presented for one particle in one dimension and is stated to generalize routinely to more particles and dimensions.
- Classically, all marginal distributions of ax + bp uniquely determine an ordinary phase-space distribution.
- Quantum measurements of z = ax + bp correspond to the Hermitian operator Z = aX + bP and provide a probability distribution for z.
- Because [X, P] = iℏI, higher commutators vanish, allowing Zassenhaus’s formula to split the exponential into manageable factors.
- Assuming correct marginals for every linear combination, the proof uses Corollary 3 and a two-dimensional inverse Fourier transform to recover f(x, p).
- The reconstructed function is Wigner’s quasi-distribution, and reversibility of the derivation verifies its correct marginal distributions.
5. Weyl’s correspondence
Weyl’s correspondence converts phase-space functions into operators through Fourier transformation and substitution of quantum operators. The paper shows that Wigner’s distribution is uniquely characterized by matching expectations for all well-behaved functions.
- Weyl’s approach characterizes Wigner’s distribution using expectation values for a broad class of functions rather than only linear marginals.
- The Weyl correspondence Fourier-transforms g(x, p) and applies the inverse transform with Hermitian operators X and P replacing x and p.
- The paper shows that Wigner’s distribution uniquely equates quantum expectations of g(X, P) with phase-space expectations of g(x, p) for all well-behaved g.
- Weyl’s Fourier-based expression provides a canonical representation of g that depends only on the function, avoiding ordering ambiguity when substituting X and P.
6. Feynman and spins
Feynman’s quasi-distribution assigns joint values to the non-commuting spin components Z and X, but its marginals alone do not determine it uniquely. Requiring correct distributions for all linear combinations instead fails because the quasi-probability and quantum-mechanical possible values differ.
- Spin quasi-distribution: Feynman models the non-commuting spin components Z and X with four quasi-probability components indexed by their ±1 values.The spin matrices are normalized to have eigenvalues ±1, and the components f++, f+−, f−+, and f−− represent joint assignments of Z and X.
- Linear combinations: For Z + X, the quasi-distribution permits values 2, 0, and −2 with probabilities f++, f+− + f−+, and f−−.These values arise by adding each jointly assigned pair of Z and X outcomes.
- Linear combinations: Quantum mechanics instead allows only the eigenvalues ±√2 for Z + X, so the quasi-distribution cannot reproduce the correct marginal for this combination.The same mismatch occurs for any nontrivial linear combination of Z and X.
- Marginal constraints: Correct marginals for Z and X alone leave one free parameter because the four marginal equations contain only three independent constraints.Thus the two individual spin distributions do not uniquely determine the joint quasi-distribution.
- Marginal constraints: Choosing the free parameter appropriately can make all four components nonnegative for every quantum state.The feasibility conditions reduce to −1 ≤ ⟨Z⟩ ≤ 1 and −1 ≤ ⟨X⟩ ≤ 1, which hold because both observables have eigenvalues ±1.